Height function

A height function measures arithmetic complexity rather than physical size. For a reduced rational number $a/b$, its multiplicative height is $H(a/b)=\max (|a|,|b|)$, and its logarithmic height is $h(a/b)=\log H(a/b)$. Thus $1001/1000$ is close to one in real size but has height $1001$.

For an algebraic number $\alpha$ of degree $d$, let its primitive irreducible integer polynomial have leading coefficient $a_d$ and complex roots ${\alpha }_1,\dots ,{\alpha }_d$. The Absolute logarithmic Weil height is $d^{-1}(\log |a_d|+{\sum }_i\log \max (1,|{\alpha }_i|))$. Bounding both height and degree leaves finitely many possibilities, making heights useful in Diophantine equations and finiteness arguments.